TITLE: p-adic analysis of hypergeometric functions
QUESTION [6 upvotes]: Are there any p-adic techniques that can be applied to the 2F1 hypergeometric function?
For e.g. I'm interested in which values this function converges p-adically.

REPLY [8 votes]: The Gauss hypergeometric function is the main example in the theory of p-adic differential equations. See 
K. S. Kedlaya, p-Adic Differential Equations, Cambridge University Press, 2010,
for the general theory. There were also two books by Dwork, almost completely devoted to ${}_2F_1$ (its p-adic theory is much more complicated than the classical one):
B. Dwork, Generalized hypergeometric functions. Oxford: Clarendon Press, 1990. 
B. Dwork, Lectures on p-adic differential equations, Springer, 1982.
It is easy to check local p-adic convergence for the hypergeometric series, but to study and even correctly define its analytic continuation properties one needs subtle analytic and algebraic techniques.